Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations

Abstract

A correspondence between the sextic anharmonic oscillator and a pair of thirdorder ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators with order greater than 2. In particular, links with Bender-Dunne polynomials and resonances between independent solutions are observed for certain secondorder cases, and extended to the higher-order problems.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.

ODE/IM correspondence and Bethe ansatz for affine Toda field equations

“…ĝ ∨ denotes the Langlands dual of ĝ, whose simple roots are α ∨ a . The simply-laced affine Lie algebras A (1) r , D (1) r , and E (1) r are self-dual, whereas the non-simplylaced cases obey (B (1) …”

“…Let us consider an embedding of modules as explained in [7] (see also [13]). In the case of A (1) r there is an embedding ι which acts as…”

“…The ODE/IM correspondence was proposed by Dorey, Dunning, and Tateo in [1] where they demonstrated an interesting relationship between a Schrödinger-type ordinary differential equation with anharmonic potential and the conformal limit of a certain two-dimensional quantum integrable model. It was shown that functional relations satisfied by the Stokes multipliers and spectral determinants of this ODE agree with those of the Q-operator and transfer matrix vacuum eigenvalues for an A 1 type quantum integrable system in the conformal field theory limit (see also [2]).…”

“…Lukyanov and Zamolodchikov [8] studied the ODE/IM correspondence for the massive sine(h)-Gordon model and found that spectral determinants of a modified form of the classical sinh-Gordon model coincide with the Q-functions of the quantum sine-Gordon model, the affine Toda field theory for algebra A (1) 1 . This was generalized to a relation between the classical Tzitzéica-Bullough-Dodd equation (A (2) 2 algebra) and the quantum Izergin-Korepin model in [9], and was studied for type A (1) r affine Toda theories in [10,11].…”

“…This was generalized to a relation between the classical Tzitzéica-Bullough-Dodd equation (A (2) 2 algebra) and the quantum Izergin-Korepin model in [9], and was studied for type A (1) r affine Toda theories in [10,11]. In these works it was shown that connection coefficients for subdominant solutions to the linear problem associated with the affine Toda field equation correspond to the vacuum eigenvalues of Q-operators for g-type quantum integrable models.…”